Time-domain green's function for an infinite sequentially ... - IEEE Xplore

160

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 51, NO. 2, FEBRUARY 2003

Time-Domain Green’s Function for an Infinite Sequentially Excited Periodic Planar Array of Dipoles Filippo Capolino, Member, IEEE, and Leopold B. Felsen, Life Fellow, IEEE

Abstract—The present paper is a continuation of previous explorations by the authors, aimed at gaining a basic understanding of the time domain (TD) behavior of large periodic phased (i.e., sequentially turned-on) array antenna and related configurations. Our systematic investigation of the relevant canonical TD dipole-excited Green’s functions has so far included those for infinite and truncated sequentially pulsed line periodic arrays, parameterized in terms of radiating (propagating) and nonradiating (evanescent) conical TD Floquet waves (FW) and truncation-induced TD FW-modulated tip diffractions. The present contribution extends these investigations to an infinite periodic sequentially pulsed planar array, which generates pulsed plane propagating and evanescent FW. Starting from the familiar frequency domain (FD) transformation of the linearly phased element-by-element summation synthesis into summations of propagating and evanescent FWs, we access the time domain by Fourier inversion. The inversion integrals are manipulated in a unified fashion into exact closed forms, which are parameterized ( ) ( ) by the single nondimensional quantity = 1 , where and are the excitation phase speed along a preferred phasing direction 1 in the array plane and the ambient wave speed, respectively. The present study deals with the practically relevant rapidly phased propagating case 1, reserving the more intricate slowly phased 1 regime for a future manuscript. Numerical reference data generated via element-by-element summation over the fields radiated by the individual dipoles with ultrawide band-limited excitation are compared with results obtained much more efficiently by inclusion of a few TD–FWs. Physical interpretation of the formal TD–FW solutions is obtained by recourse to asymptotics, instantaneous frequencies and wavenumbers, and related constructs. Of special interest is the demonstration that the TD–FWs emerge along “equal-delay” ellipses from the array plane; this furnishes a novel and physically appealing interpretation of the planar array TD–FW phenomenology. Index Terms—Antenna arrays, arrays, floquet waves, Green’s function, periodic structures, time-domain (TD) analysis, transient analysis. Manuscript received August 12, 2001. The work of F. Capolino was supported by the Italian Research National Council (CNR) under a Grant awarded in 1998–1999 for conducting research at Boston University. The work of L. B. Felsen was supported in part by the ODDR&E by MURI Grants ARO DAAG55-97-1-0013 and AFOSR F49620-96-1-0028, in part by the Engineering Research Centers Program of the National Science Foundation under Award No. EEC9986821, in part by the U.S.-Israel Binational Science Foundation, Jerusalem, Israel, under Grant 9900448, and in part by the Polytechnic University. F. Capolino was with the Department of Information Engineering, University of Siena, 53100 Siena, Italy. He is now with the Department of Electric and Computer Engineering, University of Houston, TX 77004 USA. L. B. Felsen is with the Department of Aerospace and Mechanical Engineering and the Department of Electrical and Computer Engineering, Boston University, Boston, MA 02215 USA, and is also with the Polytechnic University, Brooklyn, NY 11201 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2003.809092

I. INTRODUCTION

T

HIS paper represents the third in a series of prototype studies [1], [2] of the time-domain (TD) behavior of sequentially excited periodic dipole array configurations, motivated by similar investigations in the frequency domain (FD) [3]–[6], which have already been applied effectively and efficiently to finite practical array antennas [7]–[10]. Referring to the more detailed introduction in [1] for background, we proceed in Section II to the formulation of the problem in terms of the frequency-domain (FD) and time-domain (TD) fields excited by the individual phased dipole radiators. Via Poisson summation, these discretely spaced individual FD and TD sources are reexpressed collectively in terms of equivalent global periodicity-induced continuous distributions which obey the Floquet wave (FW) dispersion relation and span the entire array surface. The FD-FW and TD–FW wavefields, i.e., the Floquet plane waves, radiated by these FW-modulated aperture distributions are developed in Sections III and IV, respectively, with emphasis in the TD on the new phenomenologies exhibited by the planar array, as well as on similarities with the previously investigated TD-infinite line dipole array [1]. As previously cited in [1], the TD–FW fields are found to be expressible in new exact closed forms which reduce to known results for special choices of the problem parameters. Numerical results in Section V furnish reference data which are used for comparison with considerably more efficient FW-generated fields. Conclusions are presented in Section VI. II. STATEMENT OF THE PROBLEM A. Element-by-Element Formulation The geometry of the planar array of dipoles oriented along direction and excited by transient currents in free space the and in the is shown in Fig. 1. The period of the array is and directions, respectively. The vector electric field is directed magnetic scalar potential simply related to the which shall be used throughout. A caret (^) tags time-dependent , , quantities; boldface symbols define vector quantities; and denote unit vectors along , , and , respectively. FD and TD quantities are related by the Fourier transform pair

0018-926X/03$17.00 © 2003 IEEE

(1)

CAPOLINO AND FELSEN: TD GREEN’S FUNCTION

161

B. Collective Formulation To convert the individual element contributions in (2) into equivalent collective smooth aperture distributions, we use the Poisson sum formula in its most elementary form given by [11, pp. 117] . When applied sequentially to the double infinite series -indexed FD and TD elements in (2), Poisson of phased summation yields

(6) Fig. 1. Planar periodic array of dipoles. Physical configuration and coordinates. d , d : interelement spacing along x and x , respectively; ! = k : phase gradient of the excitation (i.e., the wavefront) along the = =c: “slowness” (normalized wavenumber) along i ; direction i ; v = c= = 1= : phase speed along i .

The phased array FD and TD dipole currents respectively, are given by

and

,

The vector wavenumber (7) (8) which combines the two Floquet-type dispersion relations (9) (10)

(2) (3) where ambient wavenumber, and and Moreover,

is the

th dipole location, , denotes the denotes the ambient wave speed. , with (4)

and , respecare the interelement phase gradients along of the importively. Here, is chosen to match the form tant nondimensional parameter introduced previously [1]. This yields (5)

, has previously been employed in with the FD studies of planar dipole arrays [3]–[5]. The subscript denotes the vector component transverse to , and “ ” on represents the -independent part of the vector dispersion relation in (7). Thus, in the frequency domain, Poisson summation converts the effect of the infinite periodic array of indi-indexeddipole radiators collectively into an vidual phased -indexed infinite superposition of linearly smoothly phased equivalent planar distributions that furnish the initial conditions for propagating (i.e., radiating) PFW and evanescent (i.e., non-inradiating) EFW Floquet-type waves. In the TD, the dexed sequentially pulsed dipoles are converted collectively into -indexed impulsive source distributions smoothly phased, , which travel with phase speed in the direction, which is the direction of the wavefront shown in Fig. 1. III. FLOQUET WAVES: FREQUENCY DOMAIN

now denoting the normalized (with with the corresponding respect to ) phase gradient, and direction (see impressed phase speed, along the array in the is rotated through the Fig. 1). The FD phasing unit vector with respect the axis; this corresponds in the TD angle to sequentially pulsed dipole elements, with the element at turned on at time . Choosing the noras in (5) systematizes subsequent malized form for along notation and interpretation. and characterize two disThe respective regimes along tinct TD wave phenomenologies with phase speeds larger or smaller than the ambient wave speed . Only the practiregime is examined in this paper. cally more important

To obtain for the potential fields radiated by the an equivalinearly phased dipole array element currents at radiated by the smoothly lent sum of FW potentials phased FW-modulated aperture distributions, we multiply the FD portion of (6) by the FD element Green’s function (11) 1 and 2, to generate on the left-hand denotes the position vector. side (LHS) of (6). Here, On the right-hand side (RHS) of (6), this yields the collective

and perform the integration

, for

162


FW-phased plane waves (it is the inverse of the transform shown in [12, p. 481])

with rewritten as

given in (12). The

wavenumber is

(17) (12) Here, vector, and

denotes the total

in which we used the frequency shift (18)

propagation and the definitions (13)

(19)

is the wavenumber along . The square root function in (13) on the top Riemann sheet, is defined so that . Furtherconsistent with the radiation condition at or for or 0, respectively, more, in order to satisfy the radiation condition for positive and negative real frequencies. In (12), Floquet waves with transverse or , where propagation constants , characterize PFW or EFW, respectively, along . Note that by phase matching along , , each a ray PFW contributes at the observation point asymptotic field originating at a point on the -plane. The ray emanating from the point lie from the -axis, and azon a ray with angular displacement from the axis (see Fig. 1), imuthal displacement

(20)

(14) or , for positive or negative frequencies. For and give rise to two different propagation anapproaches , the polar gles. When tends to . Beyond that limit, when , angle the polar angle becomes complex and the field becomes evanes, i.e., , defining the th cent along , with FW cutoff condition. Owing to the exponential attenuation of along , the EFW portion of converges rapidly away from the array plane and a few terms may suffice for an adequate approximation of the total radiated field.

Thus, (15) becomes

(21) The integrand has branch points at

, with [from (20)] (22)

as shown in Fig. 2. Cuts are determined by imposing on the top -plane Riemann sheet, and or 0 for or , in (13). In respectively, in accord with the definition of the -plane, the vertical dashed line at separates positive and negative frequencies (only is shown in the figure). Defining (23) Equation (21) is rewritten as

IV. FLOQUET WAVES: TIME DOMAIN

(24)

Three distinct approaches are analyzed, each describing different aspects of TD–FWs. The first two lead to exact expressions for TD–FWs, while the third leads to an asymptotic description of the same phenomena. A. Fourier Inversion From the FD The TD Floquet Wave is obtained through Fourier inversion from the frequency domain

(15) (16)

and or for or , respectively, in accord with the definitions for in (13). The positive–negative transition at occurs between the two branch points. The indentation of the integration path in (24) is chosen in accord with the radiation (causality) for any ; therefore, the integration condition at to is shifted below the branch cuts (see path from or for or Fig. 2), where in accord with the radiation condition specified in the text after (13) (see also [12, p. 35] where, to ensure the existence of the Fourier pair in (1), the variable and therefore the contour of integration in (24) is shifted slightly below the real axis into ).

with


163

function in (26) vanishes. Since

, we have

which agrees exactly with the real field radiated by an impulsively excited smooth infinite . plane source with phasing specified by B. Spatial Synthesis of TD–FWs Via Poisson Summation

Fig. 2. Topology of the complex ! -plane. Branch points are located at ! = ! . The vertical dotted line at ! = ! (! = 0) separates positive and negative ! frequencies (here, ! > 0 for simplicity). The dashed region denotes the side of the cuts where e(! ! ) > 0, according to the choice of the root for k in (13). For Fourier inversion (see (1)), the integration path is moved to the real axis and indented accordingly with respect to the singularities.

6

t , contributions arrive at the observer simultaneously from points whose locus is a distinct “equal delay” ellipse (see also, Fig. 4).

0

on the parameters , and . We now explore the behavior of -parameter ranges. the solutions for the FW for various Equation (35) describes an ellipse with foci , with and , 2, center and , as shown in Fig. 4. At the turn on time axis ratio , the foci coincide and the ellipse reduces to a point at . At later time instants, the moving along the ellipse becomes larger, with the focus direction and the focus moving toward the origin . For , the ellipse degenerates into a circle the nonphased case . When approaching cutoff with center fixed at , the axis ratio tends to infinity, and the two foci as well move to . (For , which correas the launch point in the FD, the equal delay contours sponds to evanescent becomehyperbolas(see(35)),ofwhichonlytheright-handbranch is relevant. Whether TD radiation is now possible under special phase-matched conditionsremains to be exploredfurther.) The -integral in (33) is exactly like that in [1] for a line , the -dependent -values that array of dipoles. For satisfy (35) are [1, Eq. (13)]

(37)

(a)

(b)

(c) Fig. 4. Various equal delay ellipse configurations, represented in the (u ; u ) plane at three time instants t > t > t , all greater than the turn-on time t . The signal arriving at the observer at t is generated earlier at the point x (t ) which, in the (u ; u ) plane, is located at ( z(1 ) ; 0). (a) For = 0, ellipses degenerate to circles. (b) For a generic 0 < < 1, the equal delay ellipses have axis ratio equal to 1=(1 ), and foci at u =

0 0

0

0c(1 0 ) + (01) 0 , i = 1, 2, that tend to u ! 0 ! 01, when ! 1. (c) When approaches the cutoff condition ( = 1) the axis ratio tends to infinity, and the foci as well as the launch point x (t ) approach u = 01. and u

. The two real solutions of (37) with coincide at time which, at the for observer, corresponds to the causal (wavefront) arrival time of a signal due to a smoothly phased infinite line current along , located , with launch point in the plane at at . In the moving coordinate system represents the signal along the excitation wavefront, arrival delay that the moving observer encounters with respect (Fig. 3). to the exciting current impulse located at , these solutions separate according to (37) and For and (Fig. 4). move toward , the two solutions are conjugate complex and For do not lie on the integration path; thus the -integral in (33) (causality). The two causal contribuvanishes for and are determined by tions corresponding to when using the formula , as in [1, Sec. V-A]. Substituting for from (34) and (37) and simplifying, one obtains what we shall -indexed TD–FW refer to as the causal solution for the

Squaring and rearranging (34) leads to (35) (36) and in accord with in (13) (since ; see (7) -plane, (35) defines -independent and (13)). In the plane, whose shape depends “equal delay” curves in the with

(38)


165

in which is defined in (19), and where or 0 for or , respectively. The function delimits the integration , domain to points with . Therefore, the ini.e., between . Changing tegral is nonvanishing only in the case of , and adding the variables to to the contributions gives

range arise from the stationary (saddle) points of , de. (For , is -infined by dependent and therefore not amenable to saddle point approxi, or , the real solutions yield the local mation.) For instantaneous frequencies (see Appendix B)

(43)

(39) and After expressing noting that the the odd part of the integrand does not contribute, we recall the formula , with denoting the Bessel function of zeroth-order [14, p. 28]; observing , we obtain that

defined in (16) and (23), respectively. Positive with and and , reand negative frequencies are denoted by spectively. This expression agrees with that obtained via the oppereration formed directly on the time-dependent phase in (26), after replacing the Bessel function by its large argument asymptotic approximation. in (43) The two instantaneous frequencies of the at a given point and a given instant ( in the moving reference system; see (31)) are real in the causal domain , increase with mode indexes but decrease with time , and approach their observer-independent , (defined by ) (see cutoff frequency when Fig. 7)

(44) characterize The instantaneous saddle point frequencies corresponding instantaneous wavenumbers pertaining to the observer located at at time . For specified , one obtains

(40) is defined in (22). This result, obtained by apin which plying the Poisson summation formula directly to the TD element-by-element field representation, is coincident with that in (26) obtained from the direct Fourier inversion of the FD-FW. The remarks after (26), concerning the “physically observable” TD–FW, apply here as well. C. Asymptotic Inversion From the FD 1) Local Frequencies and Wavenumbers: The behavior of the high-frequency asymptotic evaluation of the FD inversion integral in (1) (41) provides additional insight and parameterizes the TD–FW dispersion process. The manipulations here are 2-D generalizations of those carried out in [1] for the line dipole array, and the principal steps are given below. Referring to the last expression for in (12), accounts for the slowly varying amplitude terms in the integrand. The phase is given by

(45) (46)

(47) is calculated using (68). From where the wavenumber (14), the corresponding local FW propagation angles denoted and , respectively, become for specified by

(48)

(49) Evaluation of the th TD–FW integral in (41) via the standard asymptotic formula [12, p. 382] (50)

(42) and defined in (7) and (13), respectively. with The dominant contributions to the integral in the high-frequency

(51)

166


yields

(52)

where we have used (43), (47) and . arises because real saddle point The unit step function are restricted to . Combining frequencies with yields

Fig. 5. All TD–FWs propagate simultaneously toward the observer with group velocity v (t), v (t) = c. The emergence point x (t) of each TD{FW is located on the t-dependent “equal delay” ellipse defined in (31).

(53) The FD-inverted asymptotic TD–FW in (53) is the asymptotic version of the exact TD–FW in (26) obtained through Poisson summation directly in the time domain, as can be seen from the asymptotic approximation of the Bessel functions . Thus, all interpretations relating to (26) apply. The fact that all TD–FW propagate simultaneously toward the observer is in accord with the in (47), that is real when instantaneous wavenumber for all . Indeed, the asymptotic are such that frequencies whence, after turn-on , in (47) is real (con, dition for propagation). At the turn-on time and . For we have we have (see (44)); thus, the , with wavenumbers , as in the text after (14). 2) The Nondimensional Estimator: In order to assess the accuracy of the asymptotics in (50), we use the nondimensional estimator defined as [15]

6. Normalized Rayleigh pulse and its FD spectrum.

3) Group Velocity: The group velocity, which specifies the direction and propagation speed of the energy flux of the wave field, is defined as

with

Thus,

(54)

which combines the various critical problem parameters and variables. We have noted here that . The range of validity of the asymptotic solution is expressed thereby through , with the limits given by . the condition As a function of , this eliminates the near-wavefront regime and the late-time regime , for both of which . However, the validity of the asymptotic result is exwhen the dipoles are excited by a band-limited tended to waveform (see Section V-B).

, and . Inserting the instantaneous wavenumbers from (46) and in (47) and using the instantaneous propagation angle (48), yields

(55) Thus, all instantaneous

fields propagate toward the

as shown in Fig. 5. observer with group speed 4) Instantaneous Localization: To complete the connection between the exact and asymptotic results, we show that at the


167

Fig. 7. Local instantaneous radian frequency of oscillation ! (t) of the TD–FWs, evaluated at (x ; x ; z ) = (0; 0; 15d ), versus normalized time t=T , with T = d =c. Only those with p ; q 2 are shown. Parameters: d = d ; = 0:2, i = i . At turn-on t = t , ! (t) for all p; q . At the RHS, pulse-excitation spectra are shown for the two cases analyzed in Fig. 8 (! = c=d ; = 2d ) and in Fig. 10 (! = 4c=d ; = d =2). In each case, only those TD–FWs with local instantaneous frequency lying within the excitation spectrum are excited.

j jj j

j

observation point , localization through local instantaand in neous frequencies and wavenumbers (45) and (46), respectively, define localized “emergence” points (56) on the array plane; these points all lie on the -instantaneous “equal delay” ellipse defined in (31), as shown in Fig. 5. We first in (11), and thus recall the definition of

j!1

planar array. The pulse excitation function is represented as with spectrum . Accordingly, the factor multiplying in (2) becomes for the for the TD dipole FD dipole currents and currents. of the planar array is then The total BL response obtained by convolving the total TD impulse response in (58) , yielding with the BL signal

(see Fig. 5) which, when inserted together with (56) into (31), leads to (59)

(57) This identity is verified from (45)–(47) and (43), and it represents the equation of the “equal delay” ellipse in terms of instantaneous wavenumbers. In summary, at each time , all TD–FWs propagate toward the observer along -dependent , , with the same group cones, from directions velocity . These TD–FW emerge earlier from points located on the “equal delay” ellipse at time . D. The Total Physically Observable Radiated Field The total “physically observable” field radiated by the array , paired TD–FWs, is expressed as a sum of

(60) due to the planar The BL Floquet-modulated signal array can be calculated either by convolution with the exact TD–FW or by inversion of FD asymptotics. A. Convolution With the Exact FW Here, the exact FW field in (26) or (40) is used in (60). Again, , pairing defines “physisically observthe able” BL–TD–FW, yielding the real field , that also demonstrates (59) for BL excitation. B. Band-Limited Asymptotics

(58) is given by (26) or (40). The terms where the th TD–FW in the series on the RHS of (58) can also be rearranged so as to include only positive (and zero) , indexes. V. BAND-LIMITED PULSE EXCITATION We now analyze the effects of physically realizable band-limited (BL) pulsed dipole excitation on the field radiated by the

th BL field Avoiding the convolution in (60), the can be calculated as the inverse Fourier transform . Therefore, for or , using the of can be high-frequency asymptotics in Section IV-C, evaluated approximately by including the pulse spectrum in the inversion integral (41). For these short pulses, can be considered slowly varying with respect to the phase in the integrand of (41) [2], [16], and can therefore be approximated

168


by its value at the saddle point frequencies Thus, near the wavefronts

,

, 2.

(61) with (62) and ,

approximated asymptotically as in (52). Again, pairing synthesizes the real BL asymptotic

solution (63) since

,

[see (43)],

. For , which is not and amenable to -domain saddle point asymptotics (see Secis calculated by the tion IV-C), the pulsed response convolution in (60). Although the impulsively excited asymptotic wavefields in Section IV-C are valid only for early times close to (behind) the wavefronts, convolution with a waveform may enlarge the range having a band-limited spectrum of validity to later observation times behind the wavefront. For or , the relevant fields are those with in the signal bandwidth. C. Illustrative Examples To check the accuracy of the TD–FW-based BL Green’s function algorithm for the impulse-excited planar phased dipole array, we have implemented two numerical examples (see Figs. 8 and 10). The TD asymptotic solution (59), with (61), is compared there with a reference solution obtained via element-by-element summation over the pulsed radiation from all dipoles, i.e.,

(64) -series has been truncated when contributions from The . the far elements are negligible, i.e., when The chosen BL excitation is a normalized Rayleigh pulse (i.e., ) [17], with FD spectrum and central radian frequency , shown in Fig. 6. To explain the results in Figs. 8 and 10, we shall utilize plots of the TD–FW instantaneous frequency dispersions shown in Fig. 7. The relevant spectral range of that contributes significantly to the total radiated field at the observer can be assessed from Fig. 7 which shows on the left the instantaneous radian frequency trajectories for , evaluated at , and , with ; the array plotted versus normalized time , ( in (14)), and parameters are . At turn-on , all instantaneous frequencies with or . It is also seen that for , defined in (44). The index , 2 tags -frequencies, respectively. The RHS of negative/positive

Fig. 8. (x

Field radiated by an infinite planar array of dipoles observed at

; x ; z) = (0; 0; 15d ). The fields are plotted versus normalized time t=T , with T = d =c. Parameters: d = d ; = 0:2, i = i , = 2d , ! = =T .

Fig. 7, shows the pulse-excitation frequency spectra for the two ; (Fig. 8) cases under consideration: ; (Fig. 10). In each case, only and within the those TD–FWs with instantaneous frequencies are relevant, as stated in (61),(62). excitation spectrum , The fields are likewise plotted versus normalized time . Fig. 8 shows the field radiated by the array with , and , observed at with parameters , . The central radian frequency is chosen , with central wavelength as ; this implies from Fig. 7 that the for only those lie in the region where is nonTD–FWs with vanishing, and therefore furnish the dominant contributions. Indeed, in Fig. 8, excellent agreement with the reference solution has been obtained by retaining only the asymptotic terms , thereby demonstrating good convergence of the TD–FW field representation. Since the median wavelength is , larger than the interelement spacings the main feature of the pulse shape in Fig. 8, is contributed by the integrated excitation waveform of Fig. 6 and represents the , which is evaluated by the convolution in (60) [see text after (63)]. The tail after the wavefront is due to the higher , which oscillate at their distinct order FWs with , , 2, local instantaneous frequencies and thereby form the noted interference pattern. The quality of the asymptotic results in Fig. 8 up to (and beyond) is assessed by the behavior of the nondimensional in (54), as shown in Fig. 9 for . estimators is not included since it is not The estimator for amenable to saddle point asymptotics as noted in Section IV-C. for all except near turn-on In the plotted range, where . Near , the local instantaneous frequencies tend to infinity, but due to the bandin Fig. 6, TD–FWs limited excitation frequency spectrum are not excited there. with or Fig. 10 shows plots for an infinite planar array under the same conditions as in Fig. 8, except that the central radian fre. quency is now to This changes the relevant spectral range of


169

j jj j

Fig. 9. Nondimensional estimators E (t) in (54), shown for p ; q 2. The quality of the asymptotics for the pq th TD–FW is assessed by how well each 1, with respect to an arbitrarily set reference satisfies the condition E level.

bear strong notational resemblance to the FD planar array sector geometry in [5], and strong phenomenological resemblance to that of the TD infinite line dipole array in [1]. To highlight these analogies, we have used phrasings similar to those in [1] and [2] for similar concepts and methodologies. As in [1], the present prototype problem is sufficiently simple to yield the exact closed form TD solutions in (28) for Floquet-type dispersive wave phenomena, which are dispersive TD–FW radiating plane waves for ; the nonradiating case will be presented separately. The most interesting and new finding here is the exci) of the TD–FWs along “equal tation mechanism (for delay” ellipses in the array plane (see (35) and Figs. 3 and 4), and the appealing physical interpretations that follow from it (see Fig. 5). The next prototype studies will be of the TD–GFs for a semiinfinite planar [18], and thereafter for a plane-sectoral, phased dipole array. This will furnish the tools for analyzing actual finite planar arrays under short pulse conditions. The practical utility of FW-based dipole GFs for finite planar phased arrays has already been demonstrated in the frequency domain [3]–[10], and application of its TD counterpart will be guided by these FD studies. APPENDIX A DETAILS PERTAINING TO (25) in the We perform the change of variable integral in (25). Along the paths and in Fig. 2, the variable assumes real values from to , and the inverse is defined with , 2 when is on , as respectively. The integral in (25) is thus rewritten as

Fig. 10. (x

Field radiated by an infinite planar array of dipoles observed at

; x ; z ) = (0; 0; 15d ). The fields are plotted versus normalized time t=T , with T = d =c. Parameters: d = d ; = 0:2, i = i , = d =2, ! = 4=T .

, as can be seen from Fig. 7. Accordingly, it is noted that excellent agreement with the reference solution has been obtained also in this case by retaining the relevant asymptotic , thereby again demonstrating good converterms gence of the TD–FW field representation. At these shorter wavelengths, features of individual element arrivals become more pronounced but are well synthesized by a correspondingly larger number of TD–FWs.

(65) which has been recognized as a zeroth-order Hankel function of or , respectively [12, the first or second kind when p. 493], thereby establishing (25). APPENDIX B DETAILS PERTAINING TO (43) Recalling (42), the saddle point condition

VI. CONCLUSION In this paper we have extended previous studies of periodicity-induced impulsive Green’s functions for phased arrays of dipoles from the line dipole array (infinite [1] and truncated [2]) to an infinite planar array. From the detailed analyses in [1] and [2], we have gained substantial insight into relevant techniques for quantifying and interpreting TD periodicity-induced global phenomena in terms of TD–FW wavefields. The new features introduced by the assembly of an infinite periodic array in terms of phased line dipole arrays in Section IV-B therefore

or for or Note that in order to satisfy the radiation condition at Squaring and rearranging yields, using (13)

is

(66) 0, respectively, for all .

(67) with

. After recalling that , (67) has the two -indexed solu-

170


tions in (43). To sort out the correct solution for , we substitute (43) into the original (66) [recalling(13)] to obtain

[18] F. Capolino and L. B. Felsen, “Short-pulse radiation by a sequentially excited semi-infinite periodic planar array of dipoles,” in Radio ScienceInvited Paper, Ser. Special Issue 2001 URSI Int. Symp. Electromagn. Theory, Mar.-Apr. 2003.

(68) in (43) are obtained only for . Since, , the sign of depends on through the sign of the second term inside the parentheses in (43), i.e., . Since , . Thus, both the LHS we have and opposite sign and RHS of (68) have the same sign for , for , 2. This means that for both for and are real solutions of (66), while neither is a solution . for negative Real values of

REFERENCES [1] L. B. Felsen and F. Capolino, “Time domain Green’s function for an infinite sequentially excited periodic line array of dipoles,” IEEE Trans. Antennas Propagat., vol. 48, pp. 921–931, June 2000. [2] F. Capolino and L. B. Felsen, “Frequency and time domain Green’s functions for a phased semi-infinite periodic line array of dipoles,” IEEE Trans. Antennas Propagat., vol. 50, pp. 31–41, Jan. 2002. [3] F. Capolino, M. Albani, S. Maci, and L. B. Felsen, “Frequency domain Green’s function for a planar periodic semi-infinite dipole array. Part I: Truncated Floquet wave formulation,” IEEE Trans. Antennas Propagat., vol. 48, pp. 67–74, Jan. 2000. , “Frequency domain Green’s function for a planar periodic semi[4] infinite dipole array. Part II: Phenomenology of diffracted waves,” IEEE Trans. Antennas Propagat., vol. 48, pp. 75–85, Jan. 2000. [5] F. Capolino, S. Maci, and L. B. Felsen, “Asymptotic high-frequency Green’s function for a planar phased sectoral array of dipoles,” in Proc. Radio Science-Invited Paper, vol. 35, Ser. Special Issue 1998 URSI Int. Symp. Electromagn. Theory, Mar.–Apr. 2000, pp. 579–593. [6] S. Maci, F. Capolino, and L. B. Felsen, “Three dimensional Green’s function for planar rectangular phased dipole arrays,” in Proc. Wave Motion-Invited Paper, vol. 34, Special Issue on Electrodynamics in Complex Environments, Sept. 2001, pp. 263–279. [7] A. Neto, S. Maci, G. Vecchi, and M. Sabbadini, “Truncated Floquet wave diffraction method for the full wave analysis of large phased arrays. Part I: Basic principles and 2D case,” IEEE Trans. Antennas Propagat., vol. 48, pp. 594–600, Apr. 2000. , “Truncated Floquet wave diffraction method for the full wave [8] analysis of large phased arrays. Part II: Generalization to the 3D case,” IEEE Trans. Antennas Propagat., vol. 48, pp. 600–611, Apr. 2000. [9] A. Cucini, M. Albani, and S. Maci, “Truncated Floquet wave full wave (T(FW)2) analysis of large periodic arrays of rectangular waveguides, in print for,” IEEE Trans. Antennas Propagat., vol. 50, 2002. [10] F. Capolino, S. Maci, M. Sabbadini, and L. B. Felsen, “Large phased arrays on complex platforms,” in Proc. Int. Conf. Electromagnetics and Advanced Appl. (ICEAA), Torino, Italy, Sept. 10–14, 2001. [11] A. Papoulis, Systems and Transforms With Application in Optics. Malabar, FL: R.E. Krieger, 1981. [12] L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. Englewood Cliffs, NJ: Prentice-Hall, 1973. [13] S. R. Deans, The Radon Transform and Some of Its Applications. Malabar, FL: R. E. Krieger, 1993. [14] A. Erdélyi et al., Tables of Integral Transforms. New York, Canada, England: McGraw-Hill, 1954, vol. I. [15] L. B. Felsen, “Complexity architecture, phase space dynamics and problem-matched Green’s functions,” in Wave Motion-Invited Paper, vol. 34, Special Issue on Electrodynamics in Complex Environments, Sept. 2001, pp. 243–262. [16] L. B. Felsen and L. Carin, “Diffraction theory of frequency- and timedomain scattering by weakly aperiodic truncated thin-wire gratings,” J. Opt. Soc. Amer. A, vol. 11, no. 4, pp. 1291–1306, Apr. 1994. [17] P. Hubral and M. Tygel, “Analysis of the rayleigh pulse,” Geophysics, vol. 54, no. 5, pp. 654–658, May 1989.

Filippo Capolino (S’94–M’97) was born in Florence, Italy, in 1967. He received the Laurea degree (cum laude) in electronic engineering and the Ph.D. degree, from the University of Florence, Italy, in 1993 and 1997, respectively. From 1994 to 2000, he was been a lecturer of antennas at the Diploma di Laurea, University of Siena, Italy, where he was also a Research Associate. From 1997 to 1998, he was a Fulbright Research Visitor with the Department of Aerospace and Mechanical Engineering, Boston University, Boston, MA, where he continued his research with a Grant from the Italian National Council for Research (CNR), from 1998 to 1999. He is currently a research associate visiting professor with the Department of Electrical and Computer Engineering, University of Houston, Houston, TX. His research interests include theoretical and applied electromagnetics focused on high-frequency short-pulse radiation, array antennas, periodic structures, and numerical modeling. Dr. Capolino was awarded with a MMET’94 Student Paper Competition in 1994, the Raj Mittra Travel Grant for Young Scientists in 1996, the “Barzilai” prize for the best paper at the National Italian Congress of Electromagnetism (XI RiNEm) in 1996, and a Young Scientist Award for participating at the URSI Int. Symp. Electromagn. Theory in 1998. He received the R. W. P. King Prize Paper Award from the IEEE Antennas and Propagation Society for the Best Paper of the Year 2000, by an author under 36 years old.

Leopold B. Felsen (S’47–M’54–SM’55–F’62– LF’90) was born in Munich, Germany, on May 7, 1924. He received the B.E.E., M.E.E., and D.E.E. degrees from the Polytechnic Institute of Brooklyn, Brooklyn, NY, in 1948, 1950, and 1952, respectively. He emigrated to the United States in 1939 and served in the U.S. Army from 1943 to 1946. After 1952, he remained with the Polytechnic (now Polytechnic University), gaining the position of University Professor in 1978. From 1974 to 1978, he was Dean of Engineering. In 1994, he resigned from the full-time Polytechnic faculty and was granted the status of University Professor of Emeritus. He is now Professor of Aerospace and Mechanical Engineering and Professor of Electrical and Computer Engineering at Boston University, Boston, MA. He is the author or coauthor of more than 300 papers and of several books, including the classic Radiation and Scattering of Waves (Piscataway, NJ: IEEE Press, 1994). He is an Associate Editor of several professional journals and an editor of the Wave Phenomena Series (New York: Springer-Verlag). His research interests encompass wave propagation and diffraction in complex environments and in various disciplines, high-frequency asymptotic and short-pulse techniques, and phase-space methods with an emphasis on wave-oriented data processing and imaging. Dr. Felsen is a Member of Sigma Xi and a Fellow of the Optical Society of America and the Acoustical Society of America. In 1974, he was an IEEE IAPS (Antennas and Propagation Society) Distinguished Lecturer. He has held Visiting Professorships and Fellowships at universities in the United States and abroad, including the Guggenheim in 1973 and the Humboldt Foundation Senior Scientist Award in 1981. He was awarded the Balthasar van der Pol Gold Medal from the International Union of Radio Science (URSI) in 1975, an honorary doctorate from the Technical University of Denmark in 1979, the IEEE Heinrich Hertz Gold Medal for 1991, the APS Distinguished Achievement Award for 1998, the IEEE Third Millenium Medal in 2000 (nomination by APS), three Distinguished Faculty Alumnus Awards from Polytechnic University, and an IEEE Centennial Medal in 1984. Also, awards have been bestowed on several papers authored or coauthored by him. In 1977, he was elected to the National Academy of Engineering. He has served on the APS Administrative Committee from 1963 to 1966, and as Vice Chairman and Chairman for both the United States (1966–1973) and the International (1978–1984) URSI Commission B.

Time-domain green's function for an infinite sequentially ... - IEEE Xplore

Time-domain green's function for an infinite sequentially ... - IEEE Xplore

Suggest Documents